Finite model property in weakly transitive tense logics

Authors: Minghui Ma and Qian Chen
Published in: Studia Logica 111: 217–250 (2022). DOI: https://doi.org/10.1007/s11225-022-10027-0

Abstract

The finite model property (FMP) in weakly transitive tense logics is explored. Let $ \mathbb{S}=[\mathsf{wK}_t\mathsf{4}, \mathsf{K}_t\mathsf{4}] $ be the interval of tense logics between $ \mathsf{wK}_t\mathsf{4} $ and $ \mathsf{K}_t\mathsf{4} $. We introduce the modal formula $ \mathrm{t}_0^n $ for each $ n \geq 1 $. Within the class of all weakly transitive frames, $ \mathrm{t}_0^n $ defines the class of all frames in which every cluster has at most $ n $ irreflexive points. For each $ n \geq 1 $, we define the interval $ \mathbb{S}_n = [\mathsf{wK}_t\mathsf{4T}_0^{n+1}, \mathsf{wK}_t\mathsf{4T}_0^{n}] $ which is a subset of $ \mathbb{S} $. There are $ 2^{\aleph_0} $ logics in $ \mathbb{S}_n $ lacking the FMP, and there are $ 2^{\aleph_0} $ logics in $ \mathbb{S}_n $ having the FMP. Then we explore the FMP in finitely alternative tense logics and some other intervals on them. The number of logics lacking the FMP in them is either $ 0 $ or $ 2^{ \aleph_0} $, and the number of logics having the FMP in them is either finite or $ 2^{ \aleph_0} $.